Fractions as equal parts of a whole
A chocolate bar of 10 equal pieces is shared, and a pizza is cut into 8 equal slices. When Neha eats 3 slices, we say she ate 3 out of 8 equal parts, and a fraction is exactly how we write that.
A fraction tells how many equal parts of a whole are taken. The number below the line is the denominator, the number of equal parts in the whole. The number above is the numerator, the number of parts taken. A fraction with numerator 1 is a unit fraction. The parts must be equal.
In the figure: A whole cut into 8 equal parts with 3 taken is the fraction 3/8.
Rule 1: Parts taken over equal parts — Write the number of parts taken on top and the number of equal parts in the whole below. Check first that all the parts are equal in size.
Fractions as equal parts of a whole
In the app you can move this figure with your finger.
Open in the app- Easy: A bar has 10 equal pieces and 7 are eaten: the fraction eaten is 7/10.
- Medium: A garden has 6 equal beds and 5 are planted: 5/6 is planted.
- Exam-type: A shape has 9 equal parts and 4 are shaded: 4/9 is shaded and 5/9 is not.
Rule 2: Fraction of a group — To find a fraction of a group, divide the group into as many equal sets as the denominator says, then take as many sets as the numerator says.
- Easy: 1/4 of 20 marbles: 20 ÷ 4 = 5 marbles.
- Medium: 2/3 of 18 pencils: 18 ÷ 3 = 6 and 6 × 2 = 12 pencils.
- Exam-type: 3/5 of 35 pages: 35 ÷ 5 = 7 and 7 × 3 = 21 pages.
The denominator counts all the equal parts in the whole, not the parts taken.
Counting unequal pieces as a fraction, or swapping the numerator and the denominator.
Bottom number = how many equal parts in all; top number = how many you took. Memory clue: “Denominator is Down below and Divides the whole”.
Spot the difference
| Term | Meaning | In 3/8 |
|---|---|---|
| Numerator | parts taken | 3 |
| Denominator | equal parts in the whole | 8 |
| Unit fraction | one part only | 1/8 |
| Fraction of a group | ÷ denominator, × numerator | 2/3 of 18 = 12 |
Practice questions
- 1A pizza has 8 equal slices and Jay eats 3. What fraction did he eat?
- 2What is the denominator of 5/9?
- 3A ribbon is cut into 5 equal pieces. What fraction is one piece?
- 4What is 2/5 of 30?
- 5Sneha reads 3/8 of a 40-page booklet. How many pages did she read?
- 6In a class of 36, 5/9 are girls. How many boys are there?
- 7A shop has 48 mangoes. It sells 1/3 in the morning and 1/4 in the afternoon. How many are left?
- 8A rope of 60 m is used: first 2/5 of it, then 1/3 of what remains. How much rope is left?
Show answers (try them yourself first)
- 11) 3/8
- 22) 9
- 33) 1/5
- 44) 12
- 55) 15 pages
- 66) 16 boys (20 girls)
- 77) 20 mangoes (16 + 12 = 28 sold)
- 88) 24 m (24 used, then 12 more)
Fractions on the number line
On a ruler the space between 0 and 1 is cut into equal marks, and a measuring jug has marks between its numbers. A fraction has a place on such a scale, between whole numbers.
To mark a fraction on the number line, divide each unit length into as many equal parts as the denominator, then jump from 0 as many parts as the numerator. Fractions that are more than 1 lie to the right of 1.
In the figure: Each jump is 1/4, so 5/4 is 4 jumps to reach 1 and one more jump.
Rule 1: Divide, then jump — Cut every unit into denominator equal parts. The size of one jump is the unit fraction. Count the numerator jumps from 0 and mark the point.
Fractions on the number line
In the app you can move this figure with your finger.
Open in the app- Easy: 3/5: cut 0 to 1 into 5 equal parts and take 3 jumps from 0.
- Medium: 7/4: jumps of 1/4; 4 jumps reach 1 and 3 more land between 1 and 2.
- Exam-type: 6/3: each unit has 3 parts, so 6 jumps land exactly on 2.
Rule 2: Between which whole numbers? — Divide the numerator by the denominator. The quotient is the whole number just before the fraction, and the next whole number is just after it. If nothing is left over, the fraction sits exactly on a whole number.
- Easy: 17/5: 17 ÷ 5 gives 3 with 2 left, so it lies between 3 and 4.
- Medium: 29/6: 29 ÷ 6 gives 4 with 5 left, so it lies between 4 and 5.
- Exam-type: 10/5: 10 ÷ 5 = 2 with nothing left, so it sits exactly on 2.
The gaps between marks must all be equal, and the mark for 1 is the same as the fraction whose numerator and denominator match.
Counting the point 0 as the first jump, or drawing unequal gaps.
Number of parts in each unit = denominator; number of jumps = numerator. Memory clue: “Denominator Divides the gap, Numerator counts the jumps”.
Spot the difference
| Fraction | One jump | Position |
|---|---|---|
| 3/4 | 1/4 | 3 jumps, between 0 and 1 |
| 4/4 | 1/4 | exactly at 1 |
| 7/4 | 1/4 | between 1 and 2 |
| 8/4 | 1/4 | exactly at 2 |
Practice questions
- 1Into how many equal parts is each unit divided to mark 2/7?
- 2Where does 5/5 lie on the number line?
- 3Between which two whole numbers does 3/2 lie?
- 4Between which two whole numbers does 11/4 lie?
- 5Starting at 0, Mina jumps 9 times by 1/6. Between which whole numbers does she land?
- 614/7 lies exactly on which whole number?
- 7Which of 3/4, 9/10 and 5/4 is nearest to 1?
- 8Starting at 0, Rahul jumps by 1/3 each time. How many jumps does he need to reach 4?
Show answers (try them yourself first)
- 11) 7
- 22) at 1
- 33) 1 and 2
- 44) 2 and 3 (11 ÷ 4 = 2 left 3)
- 55) 1 and 2 (it is 9/6)
- 66) 2
- 77) 9/10 (only 1/10 away)
- 88) 12 jumps
Mixed fractions and fractions bigger than 1
Two and a half pizzas can also be said as 5 half-pizzas. A baker may say 2 1/2 trays or 5/2 trays and mean the same amount.
A fraction whose numerator is bigger than its denominator is more than 1. It can be written as a mixed fraction, which is a whole number together with a fraction smaller than 1, such as 2 3/4. Both forms show the same amount.
In the figure: 2 wholes and 3 quarters make 11 quarters, so 2 3/4 = 11/4.
Rule 1: Mixed fraction to a fraction — Multiply the whole number by the denominator, add the numerator, and keep the same denominator.
Mixed fractions and fractions bigger than 1
In the app you can move this figure with your finger.
Open in the app- Easy: 3 2/5: 3 × 5 = 15 and 15 + 2 = 17, so 17/5.
- Medium: 1 5/8: 1 × 8 = 8 and 8 + 5 = 13, so 13/8.
- Exam-type: 4 1/3: 4 × 3 = 12 and 12 + 1 = 13, so 13/3.
Rule 2: Fraction to a mixed fraction — Divide the numerator by the denominator. The quotient is the whole number, and the remainder over the same denominator is the fraction part.
- Easy: 17/5: 17 ÷ 5 = 3 remainder 2, so 3 2/5.
- Medium: 29/4: 29 ÷ 4 = 7 remainder 1, so 7 1/4.
- Exam-type: 24/6: 24 ÷ 6 = 4 remainder 0, so it is exactly 4.
When the remainder is 0 the fraction is a whole number, with no fraction part left.
Forgetting to multiply the whole number by the denominator, or changing the denominator.
Mixed to fraction: whole × denominator, then add the numerator. Memory clue: “Multiply, then Add” going up; “Divide, remainder on top” coming down.
Spot the difference
| Form | Other form | Working |
|---|---|---|
| 2 3/4 | 11/4 | 2 × 4 + 3 = 11 |
| 3 2/5 | 17/5 | 3 × 5 + 2 = 17 |
| 29/4 | 7 1/4 | 29 ÷ 4 = 7 remainder 1 |
| 24/6 | 4 | 24 ÷ 6 = 4 remainder 0 |
Practice questions
- 1Write 1 1/2 as a fraction.
- 2Write 7/2 as a mixed fraction.
- 3What is the whole part of 4 3/5?
- 4Write 5 3/7 as a fraction.
- 5Write 31/6 as a mixed fraction.
- 6A recipe needs 2 1/4 cups of flour and one scoop holds 1/4 cup. How many scoops are needed?
- 7How many quarters are there in 6 3/4?
- 8A rope is 47/8 m long. Write its length as a mixed fraction.
Show answers (try them yourself first)
- 11) 3/2
- 22) 3 1/2
- 33) 4
- 44) 38/7
- 55) 5 1/6
- 66) 9 scoops (2 1/4 = 9/4)
- 77) 27 quarters (6 × 4 + 3)
- 88) 5 7/8 m (47 ÷ 8 = 5 remainder 7)
Equivalent fractions and simplest form
Half a pizza is the same as 2 quarters or 4 eighths. The pizza did not change; we only cut it into smaller equal pieces, so the same amount has different names.
Equivalent fractions show the same amount. Multiply or divide the numerator and the denominator by the same number (not zero) and the value does not change. A fraction is in its simplest form when the numerator and denominator have no common factor except 1.
In the figure: 1/2, 2/4 and 4/8 have the same shaded length.
Rule 1: Multiply top and bottom — To make an equivalent fraction with bigger numbers, multiply the numerator and the denominator by the same number.
Equivalent fractions and simplest form
In the app you can move this figure with your finger.
Open in the app- Easy: 2/3 times 4 on both: 8/12.
- Medium: 5/6 times 3 on both: 15/18.
- Exam-type: 7/10 times 5 on both: 35/50.
Rule 2: Divide by the common factor — To reach the simplest form, divide the numerator and the denominator by their highest common factor. Then no common factor greater than 1 is left.
- Easy: 12/18: the highest common factor is 6, so 12 ÷ 6 = 2 and 18 ÷ 6 = 3, giving 2/3.
- Medium: 20/50: the highest common factor is 10, giving 2/5.
- Exam-type: 28/42: the highest common factor is 14, so 28 ÷ 14 = 2 and 42 ÷ 14 = 3, giving 2/3.
Doing the same multiplication or division on top and bottom keeps the value, but adding the same number to both changes it.
Adding the same number to the numerator and the denominator, or multiplying only the top.
To check whether two fractions are equal, reduce both to the simplest form and compare. Memory clue: “What you do Up, do Down”.
Spot the difference
| Task | Rule | Example |
|---|---|---|
| Bigger numbers | multiply both by the same number | 1/2 = 3/6 |
| Smaller numbers | divide both by a common factor | 6/9 = 2/3 |
| Simplest form | divide by the highest common factor | 12/18 = 2/3 |
| Check equal | reduce both | 4/6 and 6/9 are both 2/3 |
Practice questions
- 1Write an equivalent fraction of 1/2 with denominator 4.
- 2What is 4/8 in simplest form?
- 3Write 3/4 as an equivalent fraction with denominator 12.
- 4Write 18/24 in simplest form.
- 5Are 4/6 and 10/15 equivalent fractions?
- 6Find the missing number: 5/8 = ?/40.
- 7Which of 6/10, 9/15, 12/25 and 15/25 is NOT equivalent to 3/5?
- 8Write 84/126 in simplest form.
Show answers (try them yourself first)
- 11) 2/4
- 22) 1/2
- 33) 9/12
- 44) 3/4 (divide by 6)
- 55) Yes, both are 2/3
- 66) 25 (multiply by 5)
- 77) 12/25 (3/5 = 15/25)
- 88) 2/3 (divide by 42)
Comparing, adding and subtracting fractions
Two friends eat 1/2 and 1/3 of equal chapatis, and we want to know who ate more. A cook pours 3/4 litre of milk and then 1/6 litre more, and wants the total. Both need fractions to be compared and combined.
Fractions with the same denominator are like fractions. Compare them by their numerators, and add or subtract the numerators while the denominator stays. For unlike fractions, first change them to equivalent fractions with a common denominator (the LCM of the denominators), then proceed.
In the figure: 1/2 becomes 2/4, and then 2/4 + 1/4 = 3/4.
Rule 1: Compare fractions — With the same denominator, the bigger numerator is bigger. With the same numerator, the smaller denominator is bigger. Otherwise, make the denominators equal and compare the numerators.
- Easy: 5/9 and 7/9 have the same denominator, so 7/9 is bigger.
- Medium: 2/3 and 3/4: denominator 12 gives 8/12 and 9/12, so 3/4 is bigger.
- Exam-type: 3/7 and 3/5 have the same numerator, so the one with the smaller denominator, 3/5, is bigger.
Rule 2: Add and subtract — Make the denominators the same, then add or subtract only the numerators and keep the denominator. Write the answer in simplest form.
Comparing, adding and subtracting fractions
In the app you can move this figure with your finger.
Open in the app- Easy: 3/8 + 2/8 = 5/8, since the denominators already match.
- Medium: 1/2 + 1/3: denominator 6 gives 3/6 + 2/6 = 5/6.
- Exam-type: 5/6 − 1/4: denominator 12 gives 10/12 − 3/12 = 7/12.
The LCM of the denominators gives the smallest common denominator, which keeps the numbers small.
Adding the denominators as well, for example 1/2 + 1/3 = 2/5, which is wrong.
Same bottom first, then work only with the tops. Memory clue: “Bottoms Match, then work on the Tops”.
Spot the difference
| Case | Rule | Example |
|---|---|---|
| Same denominator | bigger numerator is bigger | 4/9 < 7/9 |
| Same numerator | smaller denominator is bigger | 3/5 > 3/7 |
| Add unlike | common denominator, add tops | 1/2 + 1/3 = 5/6 |
| Subtract unlike | common denominator, subtract tops | 5/6 − 1/4 = 7/12 |
Practice questions
- 1Which is bigger, 4/11 or 9/11?
- 2What is 2/9 + 5/9?
- 3What is 9/10 − 4/10 in simplest form?
- 4What is 3/4 + 1/6?
- 5What is 7/8 − 1/2?
- 6Rani uses 1/3 m of ribbon and Sita uses 1/4 m. How much ribbon do they use together?
- 7Arrange 5/6, 3/4 and 2/3 from the smallest to the largest.
- 8A tank is 2/3 full. Then 1/4 of the tank is used. What fraction of the tank is full now?
Show answers (try them yourself first)
- 11) 9/11
- 22) 7/9
- 33) 1/2 (5/10)
- 44) 11/12 (9/12 + 2/12)
- 55) 3/8 (7/8 − 4/8)
- 66) 7/12 m (4/12 + 3/12)
- 77) 2/3, 3/4, 5/6 (8/12, 9/12, 10/12)
- 88) 5/12 (8/12 − 3/12)